Problem: $ 1.\overline{95} \div 0.\overline{5} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 195.9595...\\ x &= 1.9595...\end{align*} $ $\begin{align*} 99x &= 194 \\ x &= \dfrac{194}{99}\end{align*} $ $\begin{align*} 10y &= 5.5555...\\ y &= 0.5555...\end{align*} $ $\begin{align*} 9y &= 5 \\ y &= \dfrac{5}{9}\end{align*} $ So, the problem becomes: $ \dfrac{194}{99} \div \dfrac{5}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{194}{99} \times \dfrac{9}{5} = {?} $ $ \phantom{\dfrac{194}{99} \times \dfrac{5}{9}} = \dfrac{194 \times 9}{99 \times 5} $ $ \phantom{\dfrac{194}{99} \times \dfrac{5}{9}} = \dfrac{194 \times \cancel{9}} {\cancel{99}11 \times 5} $ $ \phantom{\dfrac{194}{99} \times \dfrac{5}{9}} = \dfrac{194}{55} $